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Mirrors > Home > MPE Home > Th. List > Mathboxes > 1arymaptfv | Structured version Visualization version GIF version |
Description: The value of the mapping of unary (endo)functions. (Contributed by AV, 18-May-2024.) |
Ref | Expression |
---|---|
1arymaptfv.h | ⊢ 𝐻 = (ℎ ∈ (1-aryF 𝑋) ↦ (𝑥 ∈ 𝑋 ↦ (ℎ‘{⟨0, 𝑥⟩}))) |
Ref | Expression |
---|---|
1arymaptfv | ⊢ (𝐹 ∈ (1-aryF 𝑋) → (𝐻‘𝐹) = (𝑥 ∈ 𝑋 ↦ (𝐹‘{⟨0, 𝑥⟩}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 6818 | . . 3 ⊢ (ℎ = 𝐹 → (ℎ‘{⟨0, 𝑥⟩}) = (𝐹‘{⟨0, 𝑥⟩})) | |
2 | 1 | mpteq2dv 5191 | . 2 ⊢ (ℎ = 𝐹 → (𝑥 ∈ 𝑋 ↦ (ℎ‘{⟨0, 𝑥⟩})) = (𝑥 ∈ 𝑋 ↦ (𝐹‘{⟨0, 𝑥⟩}))) |
3 | 1arymaptfv.h | . 2 ⊢ 𝐻 = (ℎ ∈ (1-aryF 𝑋) ↦ (𝑥 ∈ 𝑋 ↦ (ℎ‘{⟨0, 𝑥⟩}))) | |
4 | eqid 2736 | . . . . 5 ⊢ (0..^1) = (0..^1) | |
5 | 4 | naryrcl 46317 | . . . 4 ⊢ (ℎ ∈ (1-aryF 𝑋) → (1 ∈ ℕ0 ∧ 𝑋 ∈ V)) |
6 | 5 | simprd 496 | . . 3 ⊢ (ℎ ∈ (1-aryF 𝑋) → 𝑋 ∈ V) |
7 | 6 | mptexd 7150 | . 2 ⊢ (ℎ ∈ (1-aryF 𝑋) → (𝑥 ∈ 𝑋 ↦ (ℎ‘{⟨0, 𝑥⟩})) ∈ V) |
8 | 2, 3, 7 | fvmpt3 6929 | 1 ⊢ (𝐹 ∈ (1-aryF 𝑋) → (𝐻‘𝐹) = (𝑥 ∈ 𝑋 ↦ (𝐹‘{⟨0, 𝑥⟩}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 Vcvv 3441 {csn 4572 ⟨cop 4578 ↦ cmpt 5172 ‘cfv 6473 (class class class)co 7329 0cc0 10964 1c1 10965 ℕ0cn0 12326 ..^cfzo 13475 -aryF cnaryf 46312 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5226 ax-sep 5240 ax-nul 5247 ax-pr 5369 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4269 df-if 4473 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4852 df-iun 4940 df-br 5090 df-opab 5152 df-mpt 5173 df-id 5512 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-ov 7332 df-oprab 7333 df-mpo 7334 df-naryf 46313 |
This theorem is referenced by: 1arymaptf1 46328 |
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